In mathematics, a representation on coordinate rings is a representation of a group on coordinate rings of affine varieties.
Let X be an affine algebraic variety over an algebraically closed field k of characteristic zero with the action of a reductive algebraic group G. G then acts on the coordinate ring
k
[
X
]
of X as a left regular representation:
(
g
⋅
f
)
(
x
)
=
f
(
g
−
1
x
)
. This is a representation of G on the coordinate ring of X.
The most basic case is when X is an affine space (that is, X is a finite-dimensional representation of G) and the coordinate ring is a polynomial ring. The most important case is when X is a symmetric variety; i.e., the quotient of G by a fixed-point subgroup of an involution.
Let
k
[
X
]
(
λ
)
be the sum of all G-submodules of
k
[
X
]
that are isomorphic to the simple module
V
λ
; it is called the
λ
-isotypic component of
k
[
X
]
. Then there is a direct sum decomposition:
k
[
X
]
=
⨁
λ
k
[
X
]
(
λ
)
where the sum runs over all simple G-modules
V
λ
. The existence of the decomposition follows, for example, from the fact that the group algebra of G is semisimple since G is reductive.
X is called multiplicity-free (or spherical variety) if every irreducible representation of G appears at most one time in the coordinate ring; i.e.,
dim
k
[
X
]
(
λ
)
≤
dim
V
λ
. For example,
G
is multiplicity-free as
G
×
G
-module. More precisely, given a closed subgroup H of G, define
ϕ
λ
:
V
λ
∗
⊗
(
V
λ
)
H
→
k
[
G
/
H
]
(
λ
)
by setting
ϕ
λ
(
α
⊗
v
)
(
g
H
)
=
⟨
α
,
g
⋅
v
⟩
and then extending
ϕ
λ
by linearity. The functions in the image of
ϕ
λ
are usually called matrix coefficients. Then there is a direct sum decomposition of
G
×
N
-modules (N the normalizer of H)
k
[
G
/
H
]
=
⨁
λ
ϕ
λ
(
V
λ
∗
⊗
(
V
λ
)
H
)
,
which is an algebraic version of the Peter–Weyl theorem (and in fact the analytic version is an immediate consequence.) Proof: let W be a simple
G
×
N
-submodules of
k
[
G
/
H
]
(
λ
)
. We can assume
V
λ
=
W
. Let
δ
1
be the linear functional of W such that
δ
1
(
w
)
=
w
(
1
)
. Then
w
(
g
H
)
=
ϕ
λ
(
δ
1
⊗
w
)
(
g
H
)
. That is, the image of
ϕ
λ
contains
k
[
G
/
H
]
(
λ
)
and the opposite inclusion holds since
ϕ
λ
is equivariant.
Let
v
λ
∈
V
λ
be a B-eigenvector and X the closure of the orbit
G
⋅
v
λ
. It is an affine variety called the highest weight vector variety by Vinberg–Popov. It is multiplicity-free.
